# Logit/Cloglog as Primary Estimator with Proper Scoring

## Base Rates

- **Banking**: 0.0081 (144 events / 17775 obs = 0.81%)
- **Any Crisis**: 0.0222 (395 events / 17775 obs = 2.22%)
- **CA Reversal**: 0.0759 (1350 events / 17775 obs = 7.59%)

## Proper Scoring Rules: Model Comparison

| Dep. Var | Model | R²/Pseudo-R² | Brier Score | Log Loss | AUC |
|:---|:---|---:|---:|---:|---:|
| Banking | Banking EW only (LPM) | 0.0037 | 0.014121 | 0.262791 | 0.619 |
| Banking | Banking EW only (Logit) | 0.0443 | 0.013918 | 0.071103 | 0.708 |
| Banking | Banking EW only (Cloglog) | 0.0438 | 0.013918 | 0.071133 | 0.707 |
| Banking | Banking EW+Z (LPM) | 0.0076 | 0.014124 | 0.283631 | 0.640 |
| Banking | Banking EW+Z (Logit) | 0.0661 | 0.013841 | 0.069480 | 0.731 |
| Banking | Banking EW+Z (Cloglog) | 0.0658 | 0.013842 | 0.069503 | 0.730 |
| Any Crisis | Any Crisis EW only (LPM) | 0.0173 | 0.034291 | 0.498104 | 0.680 |
| Any Crisis | Any Crisis EW only (Logit) | 0.0668 | 0.033101 | 0.141453 | 0.723 |
| Any Crisis | Any Crisis EW only (Cloglog) | 0.0650 | 0.033118 | 0.141731 | 0.722 |
| Any Crisis | Any Crisis EW+Z (LPM) | 0.0198 | 0.034440 | 0.575614 | 0.665 |
| Any Crisis | Any Crisis EW+Z (Logit) | 0.0771 | 0.032862 | 0.139895 | 0.728 |
| Any Crisis | Any Crisis EW+Z (Cloglog) | 0.0753 | 0.032893 | 0.140163 | 0.725 |
| CA Reversal | CA Reversal EW only (LPM) | 0.0328 | 0.154783 | 1.969783 | 0.654 |
| CA Reversal | CA Reversal EW only (Logit) | 0.0405 | 0.135176 | 0.441505 | 0.632 |
| CA Reversal | CA Reversal EW only (Cloglog) | 0.0422 | 0.135039 | 0.440748 | 0.633 |
| CA Reversal | CA Reversal EW+Z (LPM) | 0.0511 | 0.142720 | 1.151857 | 0.655 |
| CA Reversal | CA Reversal EW+Z (Logit) | 0.0591 | 0.132662 | 0.432983 | 0.659 |
| CA Reversal | CA Reversal EW+Z (Cloglog) | 0.0608 | 0.132507 | 0.432179 | 0.660 |

## Reframing: Absolute Improvement Context

### Banking (base rate = 0.0142)
- Pseudo-R² (Logit): EW-only 0.0443 → EW+Z 0.0661 (Δ = 0.0218, 49.3% improvement)
- Brier score: 0.013918 → 0.013841 (Δ = 0.000077, 0.6% reduction in prediction error)
- AUC: 0.708 → 0.731

### Any Crisis (base rate = 0.0350)
- Pseudo-R² (Logit): EW-only 0.0668 → EW+Z 0.0771 (Δ = 0.0103, 15.4% improvement)
- Brier score: 0.033101 → 0.032862 (Δ = 0.000239, 0.7% reduction in prediction error)
- AUC: 0.723 → 0.728

### CA Reversal (base rate = 0.1727)
- Pseudo-R² (Logit): EW-only 0.0405 → EW+Z 0.0591 (Δ = 0.0185, 45.7% improvement)
- Brier score: 0.135176 → 0.132662 (Δ = 0.002514, 1.9% reduction in prediction error)
- AUC: 0.632 → 0.659


## Calibration (Logit EW+Z, Predicted Probability Deciles)

### Banking
| Bin | N | Mean Predicted | Mean Observed |
|:---|---:|---:|---:|
| 0.00-0.10 | 5284 | 0.0139 | 0.0140 |
| 0.10-0.20 | 4 | 0.1162 | 0.2500 |
| 0.20-0.30 | 3 | 0.2434 | 0.0000 |
| 0.30-0.40 | 1 | 0.3208 | 0.0000 |

### Any Crisis
| Bin | N | Mean Predicted | Mean Observed |
|:---|---:|---:|---:|
| 0.00-0.10 | 5124 | 0.0309 | 0.0308 |
| 0.10-0.20 | 138 | 0.1254 | 0.1304 |
| 0.20-0.30 | 21 | 0.2295 | 0.3810 |
| 0.30-0.40 | 3 | 0.3459 | 0.3333 |
| 0.40-0.50 | 2 | 0.4117 | 0.0000 |
| 0.50-0.60 | 1 | 0.5124 | 0.0000 |
| 0.60-0.70 | 2 | 0.6568 | 0.0000 |
| 0.70-0.80 | 1 | 0.7031 | 0.0000 |

### CA Reversal
| Bin | N | Mean Predicted | Mean Observed |
|:---|---:|---:|---:|
| 0.00-0.10 | 1108 | 0.0756 | 0.0930 |
| 0.10-0.20 | 2618 | 0.1516 | 0.1356 |
| 0.20-0.30 | 1197 | 0.2363 | 0.2206 |
| 0.30-0.40 | 232 | 0.3411 | 0.4526 |
| 0.40-0.50 | 75 | 0.4368 | 0.6000 |
| 0.50-0.60 | 35 | 0.5464 | 0.7143 |
| 0.60-0.70 | 11 | 0.6300 | 0.3636 |
| 0.70-0.80 | 12 | 0.7625 | 0.7500 |
| 0.80-0.90 | 4 | 0.8251 | 1.0000 |
